\section{Results}
The algorithm we implemented in the Cognitive Engine accurately adapts to changing noise environments and remembers optimal parameters used in previously encountered environments. The user is able to modify their application to specify different utility functions. Our utility function is designed so that the CE will maximize throughput while minimizing power and bit error rate. The modulation schemes used are BPSK, QPSK, 8PSK, and 16-QAM, and the power is allowed to vary from 0.3 to 1.4 units in increments of 0.02 units. The modulation schemes are arranged and given numbers. As shown in the table, BPSK is represented by 1, QPSK is represented by 2, 8PSK is represented as 3 and 16-QAM is represented as 4. The observable used in our CE is noise standard deviation which is proportional to Signal-to-Noise Ratio.
\begin{center}
\begin{tabular}{ | l | c | }
  \hline                       
  Modulation Scheme & $m$ \\ \hline
  BPSK & 1 \\
  \hline
  QPSK & 2 \\
  \hline
  8PSK & 3 \\
  \hline
  16-QAM & 4 \\
  \hline  
\end{tabular}
\end{center}
The overall utility of the radio is calculated as $U = \frac{m}{200} + (1 - B)$ where $B$ is the bit error rate of the system. This utility function was chosen so that a user would be indifferent to exchanging a better modulation scheme for two or three bit errors out of 512 transmitted bits. To test our CE, more noise is added to the channel environment. Then the noise is greatly reduced and the processes begins again. After testing our engine, we found that the cognitive radio adapts to the increasing noise environment and finds a close-to-optimal solution in 20 cycles with 400 possible states. After the cognitive engine has revisited one of the previous channel states, it adopts the close-to-optimal solution instantly.

\begin{figure}[ht]
\centering
\includegraphics[width=9cm]{../../../Pictures/images/Screenshot.png}
\caption{Low-Noise Solution: 16-QAM}
\label{fig:subfig1}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=9cm]{../../../Pictures/images/Screenshot-2.png}
\caption{High-Noise Solution: BPSK}
\label{fig:subfig2}
\end{figure}

Figures ~\ref{fig:subfig1} and ~\ref{fig:subfig2} show results from our implementation of this system. When the noise standard deviation increases, the CE adopts a modulation type that is more noise immune. When the noise standard deviation is much lower, then the CE adopts a modulation that favors increased throughput. 